What is the integral of 6xsec^2(x^2)cos(tan(x^2)) ?

The integral of 6xsec^2(x^2)cos(tan(x^2)) is 3sin(tan(x^2))+C

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integral of 6xsec^2(x^2)cos(tan(x^2))

\int 6 x sec^{2} (x^{2}) cos (tan (x^{2})) d x

3 sin (tan (x^{2})) + C

Solution steps

\int 6 x sec^{2} (x^{2}) cos (tan (x^{2})) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int x sec^{2} (x^{2}) cos (tan (x^{2})) d x

Apply u-substitution

: \int \frac{sec ^{2} ( u ) cos ( tan ( u ) )}{2} d u

= 6 \cdot \int \frac{sec ^{2} ( u ) cos ( tan ( u ) )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \frac{1}{2} \cdot \int sec^{2} (u) cos (tan (u)) d u

Apply u-substitution

: \int cos (v) d v

= 6 \cdot \frac{1}{2} \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= 6 \cdot \frac{1}{2} sin (v)

Substitute back

= 6 \cdot \frac{1}{2} sin (tan (x^{2}))

Simplify

6 \cdot \frac{1}{2} sin (tan (x^{2})) : 3 sin (tan (x^{2}))

= 3 sin (tan (x^{2}))

Add a constant to the solution

= 3 sin (tan (x^{2})) + C

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What is the integral of 6xsec^2(x^2)cos(tan(x^2)) ?
The integral of 6xsec^2(x^2)cos(tan(x^2)) is 3sin(tan(x^2))+C